The Item Set Collection was introduced during the Markarth DLC (Update 28). You can collect all non-crafted item sets in the game and have them added to the sticker book. All you have to do is bind the item pieces to your account. That way they will get added to the collection.
It will cost you 75 transmute crystals to reconstruct an item piece if you only have 1 item piece of a certain set collected. However, if you collect all item pieces of a certain set it will only cost 25 transmute stones to reconstruct an item.
The item that you reconstruct will always have its baseline quality. That means, a Mythic item for example such as the Ring of the Wild Hunt will always be in legendary quality. Therefore all you need to reconstruct a Mythic item (with a trait of your choice) is 25 transmute stones, no addition materials such as improvement materials are needed.
Sets such as Perfected False God's Devotion or [Perfected Lokkestiiz] always drop their perfected jewelry in gold. Therefore the baseline quality is gold, so all you need is 25 transmute stone to get a gold ring or necklace with a trait of your choice, versus 50 transmute stones if you decide to transmute it.
In Cloudrest you can obtain the Perfected Mantle of Siroria, the perfected jewelry actually also drops in epic quality, therefore the baseline quality will be epic. Now you can reconstruct the item for 25 transmute stones and then upgrade it to legendary quality. Unless you have already made that in a previous update, then transmuting the legendary item for 50 transmute stones will be cheaper than reconstructing it for 25 transmute stones and gold platings.